Static and Kinematic Indeterminacy

1. Unknown Internal Forces (SI)

2. Unknown External Forces (SI)

3. Unknown Movements (KI)

1. Internal Forces: They are the forces produced in a member of the structure under various loading conditions.

Ex. Axial Force: i) Compressive/Tensile Force

Transverse Force: ii) Shear Force

iii) Bending Moment

Twisting Force (Torque): iv) Torsional Moment

2. External Forces: They are support reactions produced at the joint of the structural members under various loading conditions.

– It depends on the type of support provided at joint.

“Degree of Freedom (DoF)” is just total number of movement or freedom to move and various direction.

Type of Possible Movements for a particle in space: i) Displacement ii) Rotation

In order to define the movements, we will take the help of Cartesian System. Considering X axis, Y axis & Z axis for 3 dimensional movement as standard directions. Mathematically, combination of 3 standard directional movement can define any movement in 3 D.

So, in totality there are 6 Nos. of standard possible movements i.e. 3 Nos. of Displacement (along with X, Y & Z axis) and 3 Nos. of Rotation 9 (along with X, Y & Z axis).

Hence, only 6 Nos. of Degree of Freedom possible in this universe!!!

Coming Back to the Concept of SUPPORTS. Support is the phenomenon that restrains (does not allow) the Freedom for movement. (E.g. Fix Support restrains all 6 Degree of Freedom. Whereas a Hinge Support restrains only displacement but allows rotation.)

Since, movement due to external force is restrained by support/s, reacting forces would be produced at the point of support position. These reacting forces are called “Support Reactions”.

3. Unknown Movements

Now you know what degree of freedom is, it is easy to understand. DOF allows the movements (displacements and rotations) and due to that there will be some unknown value of Displacement and Rotation. But where support is provided, depending upon the nature of support, value of one or more Displacements/Rotations will be ZERO (a definite known value).

Kinematics Indeterminacy (KI)

KI = No. of Unknown Movements

KI = ((DOF at Each Joint) x (No. of Joints)) – (No. of Support Reactions)

Where, DOF = 6 for 3D Frames (i.e. DISPx, DISPy, DISPz, ROTx, ROTy, ROTz)

DOF = 3 for 3D Truss (i.e. DISPx, DISPy, DISPz)

DOF = 3 for 2D Frames (i.e. DISPx, DISPy, ROTz)

DOF = 2 for 2D Truss (i.e. DISPx, DISPy)

Static Indeterminacy (SI)

SI = (SI External + SI Internal) – (Nos. of Static Equation)

Equation of Static: In static (non-moving) structure, it can be inferred that there is no actual movement. It shows that external force is balanced with equivalent internal forces and support reactions making effective resultant force ZERO.

More specifically, it means value of resultant FORCES and MOMENTS in all 3 directions can be considered as ZERO.

No. of Static Equation = (DOF at Each Joint) x (No. of Joints)

Where, DOF = 6 for 3D Frames (i.e. DISPx, DISPy, DISPz, ROTx, ROTy, ROTz)

DOF = 3 for 3D Truss (i.e. DISPx, DISPy, DISPz)

DOF = 3 for 2D Frames (i.e. DISPx, DISPy, ROTz)

DOF = 2 for 2D Truss (i.e. DISPx, DISPy)

SI External = (No. of Internal Forces) x (No. of Members)

Where, IF = 6 for 3D Frames (i.e. Fx(Axial), Fy(Shear), Fz(Shear), Mx(Torsion), My(Bending), Mz (Bending))

IF = 3 for 3D Truss (i.e. Fx(Axial), Fy(Axial), Fz(Axial))

IF = 3 for 2D Frames (i.e. Fx(Axial), Fy(Shear), My(Bending))

IF = 2 for 2D Truss (i.e. Fx(Axial), Fy(Axial))

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